Diseased or otherwise deficient heart valves can be repaired or replaced using a variety of different types of heart valve surgeries. Many heart valve surgeries involve an open-heart surgical procedure that is conducted under general anesthesia, during which the heart is stopped while blood flow is controlled by a heart-lung bypass machine. In some of these surgical procedures, it is necessary to remove portions of the native heart structure, such as the aorta, in order to restore proper functioning to a particular portion of the heart. In one example, after performing an aortotomy and removing the native aortic valve leaflets, a bioprosthesis is placed inside the native aorta and sewn into the patient's anatomy as a replacement for the aortic valve. In another example, a stented heart valve is sewn into place within patient's anatomy to replace an otherwise deficient valve. However, there is a desire to provide a replacement heart valve system that does not need to be sewn into place once it has been positioned properly relative to the anatomy of the patient in order to minimize the invasiveness of the procedure and minimize the time required to complete the surgical procedure.